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25 Feb 2019, 01:08
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B
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Difficulty:
45%
(medium)
Question Stats:
67% (01:56) correct
33%
(02:10)
wrong
based on 183
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For positive integers m and n, is m^n a perfect square?
1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9
1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9
Archit3110
25 Feb 2019, 01:58
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MathRevolution
from 1 :for a no to be multiple of 4 ; it has to be divisible twice by 4 ; so m = 2,4,6,8
; in sufficient
from 2:
234n5 ; multiple of 9
sum of digits has to be = 9
here 14+n ; n = 4
now we
know that for any integer value raised to 4 ; will always be a perfect square of a no
IMO B
27 Feb 2019, 05:17
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
The question asks if n is an even integer or m is a perfect square. Condition 2):
“23,4n5 is a multiple of 9” is equivalent to the statement that 2 + 3 + 4 + n + 5 = n + 14 is a multiple of 9. For this to occur, we must have n = 4 and m^n = m4 = (m^2)^2 is a perfect square. Condition 2 is sufficient.
Condition 1)
“12,3m0 is a multiple of 4” is equivalent to the statement that m is an even integers, since this is what is required for 12,3m0 to be a multiple of 4.
Thus, condition 1) tells us that m = 0, 2, 4, 6 or 8. Since we don’t know the exponent n, condition 1) is not sufficient.
Therefore, B is the answer.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
The question asks if n is an even integer or m is a perfect square. Condition 2):
“23,4n5 is a multiple of 9” is equivalent to the statement that 2 + 3 + 4 + n + 5 = n + 14 is a multiple of 9. For this to occur, we must have n = 4 and m^n = m4 = (m^2)^2 is a perfect square. Condition 2 is sufficient.
Condition 1)
“12,3m0 is a multiple of 4” is equivalent to the statement that m is an even integers, since this is what is required for 12,3m0 to be a multiple of 4.
Thus, condition 1) tells us that m = 0, 2, 4, 6 or 8. Since we don’t know the exponent n, condition 1) is not sufficient.
Therefore, B is the answer.
